Question

The value of \(\displaystyle\iint_S (yz dydz + zxdzdx + xydxdy)\) where S is the surface of unit sphere x² + y² + z² = 1 is

Answer 1

Correct Answer - Option 1 : 0

Concept:

According to the divergence theorem,

\(\mathop{\iint }_{S}^{{}}\bar{F}\cdot \hat{n}~~ds=\iiint_{V}{\nabla \cdot \bar{F}~~dV}\)

In expanded form,

\(\mathop{\iint }_{S}^{{}} {F_xdydz}+{F_ydzdx}+{F_zdydx} =\iiint_{V}({\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}) dxdydz}\)

Calculation:

Given F_x = yz, F_y = zx, F_z = xy;

Substituting in the theorem,

\(⇒ I = \iiint_{V}({\frac{\partial (yz)}{\partial x}+\frac{\partial (zx)}{\partial y}+\frac{\partial (xy)}{\partial z}) dxdydz}\)

\(⇒ I = \iiint_{V} {0 \; dxdydz}\)

⇒ I = 0